Static Optimization

aM

tp

g

s

ed

ta

OpenSim Workshop

The Inverse Problem

Position Data

Moments Accelerations Velocities.

Musculoskeletal Geometry

Multibody Dynamics

Forces

d dt

d dt

Force Data

Angles

Inverse Dynamics

Inverse Kinematics

Static Optimization

•  Use musculoskeletal geometry and assumptions about force distribution to estimate individual muscle forces

Major flexors

aM

Tibialis posterior

Soleus

Gastrocnemius

Major extensors

Extensor digitorum

Tibialis anterior

Net ankle moment

tp

g

sed

ta

OpenSim Workshop

Key Concepts

•  Kinematics coordinates and their velocities and accelerations

•  Kinetics muscle forces •  Muscle muscle activation-contraction

physiology dynamics and force-length- velocity relations

•  Dynamics equations of motion •  Musculoskeletal muscle moment arm

geometry •  Optimization the “distribution” problem

OpenSim Workshop

Kinetics: Muscle Forces

•  Kinetics –  Muscle forces cause the model to accelerate

•  Muscle force –  Applied between origin

and insertion points

OpenSim Workshop Muscle Velocity Muscle Length

F0M

l0M V max

Forc

e Muscle Physiology: Muscle Activation-Contraction and Force-Length-Velocity Relations

•  Muscle activation-contraction –  Biochemical reaction that causes a muscle’s fibers to

contract which produces force •  Muscle force-length-velocity

–  Force production diminshes for short, long, and fast fibers

a=1.0 a=0.5

F0M0.5

V max0.5

OpenSim Workshop

Musculoskeletal Geometry: Muscle Moment Arm

•  Muscle moment arm –  The perpendicular distance from the

line of action of a muscle to the joint center of rotation

–  Transformation from linear force of muscle to angular moment about a joint center ma

xFFrmax ˆ•×

=

r

OF

1P

2P

xy

z

OpenSim Workshop

Major flexors

aM

Tibialis posterior

Soleus

Gastrocnemius

Major extensors

Extensor digitorum

Tibialis anterior

Net ankle moment

tp

g

s

ed

ta

Static Optimization

•  Determines the “best” set of muscle forces that –  Produce net joint moments at a discrete time –  Do not violate muscle force limits –  Optimize a performance criterion

•  Performance criterion attempts to capture the goal of the neural control system –  Minimize muscle force? –  Minimize muscle stress?

OpenSim Workshop

Major flexors

aM

Tibialis posterior

Soleus

Gastrocnemius

Major extensors

Extensor digitorum

Tibialis anterior

Net ankle moment

tp

g

s

ed

ta

The Muscle Force Distribution Problem

∑∑ += structuresother to due momentsmoments musclejM

∑−∑==

=ne

eee

nf

fffj rFrFM

11

flexion moment

extension moment

number of flexors

number of extensors

moment arm

1 equation with nf + ne unknowns

( ) ( )tptpssggededtataa rFrFrFrFrFM ++−+=

Ankle example

How can we solve this?

OpenSim Workshop

Major flexors

aM

Tibialis posterior

Soleus

Gastrocnemius

Major extensors

Extensor digitorum

Tibialis anterior

Net ankle moment

tp

g

s

ed

ta

Static Optimization Formulation

( )mFfminimize

subject to

Function of muscle forces

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ]trtFtrtFtrtFtrtFtrtFtM tptpssggededtataa ++−+=

( ) N900≤tFta( ) N800≤tFed( ) N1500≤tFg( ) N2500≤tFs( ) N1500≤tFtp

OpenSim Workshop

Major flexors

aM

Tibialis posterior

Soleus

Gastrocnemius

Major extensors

Extensor digitorum

Tibialis anterior

Net ankle moment

tp

g

s

ed

ta

Example Performance Criteria Difficult to define and validate a good criterion

( ) ∑=

=nm

mmm FFf

1Muscle force

( ) ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

nm

m m

mm PCSA

FFf1

3

(Muscle stress)3 ~ Metabolic energy

Possible validations •  Use output to drive a forward dynamic

simulation •  Compare qualitatively to experimental EMG •  Compare to measured forces (instrumented

hip implant, buckle transducer in tendon)

( ) ( )∑∑==

≈⎟⎟⎠

⎞⎜⎜⎝

⎛=

nm

mm

nm

m m

mm a

PCSAFkFf

1

2

1

2

(Muscle activation)2

OpenSim Workshop

Exercise

1. Given that the rectus femoris muscle has a peak isometric force of 1169 N and it is at its optimal fiber length and zero velocity, what is the force generated for an activation of 0.86?

A. 164 N B. 952 N C. 1005 N D. 1058 N

OpenSim Workshop

Exercise

2. For the model shown on the right, which muscle has the largest moment arm about the ankle joint?

A. 1 B. 2 C. Neither (are identical)

OpenSim Workshop

Exercise

3. For the model shown on the right, which muscle has the largest moment arm about the knee joint?

A. 1 B. 2 C. Neither (are identical)

OpenSim Workshop

Exercise

4.  For the model shown on the right, muscle 1 and 2 have the following properties

To solve the “distribution” problem minimizing the sum of squared activations, which muscle would be activated more for a given dorsiflexion moment? A. 1 B. 2 C. Neither (are identical)

Muscle Peak Isometric

Force (N)

Moment Arm (cm)

1 905 3.6

2 512 3.0

OpenSim Workshop

Static Optimization in OpenSim

Model

Kinematics or States

( )∑=

nm

m

pma

1

minExponent

Idealized actuators or dependent on F-L-V?

Inputs

Residual or Reserve Actuators

External Loads

Outputs Controls (.xml)

Activations (.sto)

Force (.sto)

OpenSim Workshop

Static Optimization

TIPS & TRICKS Inputs: Can use kinematics from IK or RRA. If using IK, need to !lter kinematics

Residuals: Add residual actuators to pelvis

Reserves: Add reserve torque actuators to trouble-shoot a weak model

Minimizing residuals & reserves: Increase maximum control value (default = 1) and lower the maximum force à penalizes activity

Command Line: analyze –S setup_!le.xml